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Understanding Polar Equations of Conics: Focus, Directrix, and Eccentricity

This section explores the definition of conic sections through the focus-directrix property, illustrating how all conics can be represented by a constant ratio of distances to a focus and a directrix. The eccentricity (e) determines the type of conic: hyperbola (e > 1), ellipse (e < 1), or parabola (e = 1). Additionally, it delves into the polar equations for conics derived from the eccentricity and the directrix location, presenting the formulas based on various orientations of the conic in relation to its focus and directrix.

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Understanding Polar Equations of Conics: Focus, Directrix, and Eccentricity

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  1. Section 8-5 Polar Equations of Conics

  2. Focus –Directrix Definition • all conics can be defined using a special definition (similar to the parabola definition) • a conic section is the set of all points whose distances from a particular point (focus) and a particular line (the directrix) have a constant ratio • the constant ratio is the eccentricity (e) • if e > 1 hyperbola if e < 1 ellipse if e = 1 parabola

  3. Focus Directrix Defintion • let P be a point on the conic, F is the focus, and D be the closest point on the directrix P D F the directrix

  4. Polar Equation of Conics • the new formula for eccentricity can be used to derive polar equations for conics • place the focus of the conic at (0 , 0) and call the directrix x = k P(r,θ) D F(0,0) x = k

  5. Polar Equation of Conics P(r,θ) D r θ F(0,0) rcos θ x = k

  6. Polar Equations of Conics

  7. Polar Equations of Conics • the formula derived in the previous slide is the polar form for a conic based on the eccentricity (e) and a value k which comes from the directrix • there are three other formulas that are just like it depending on the orientation of the conic in relation to the focus and directrix • the next four slides show the four different formula and their orientation

  8. F(0 , 0) directrix x = k

  9. F (0 , 0) directrix x = – k

  10. directrix y = k F(0 , 0)

  11. F(0 , 0) directrix y = – k

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